Homological Invariants of Monomial and Binomial Ideals By Neelakandhan Manoj Kummini Submitted to the Department of Mathematics and the Faculty of the Graduate School of the University of Kansas in partial fulfillment of the requirements for the degree of Doctor of Philosophy Craig Huneke, Chairperson Perry Alexander Committee members Margaret Bayer Daniel Katz Jeremy Martin Date defended: The Dissertation Committee for Neelakandhan Manoj Kummini certifies that this is the approved version of the following dissertation: Homological Invariants of Monomial and Binomial Ideals Committee: Craig Huneke, Chairperson Perry Alexander Margaret Bayer Daniel Katz Jeremy Martin Date approved: 2 Abstract HOMOLOGICAL INVARIANTS OF MONOMIAL AND BINOMIAL IDEALS Neelakandhan Manoj Kummini The University of Kansas. Advisor: Craig Huneke. August 2008. In this dissertation, we study numerical invariants of minimal graded free resolu- tions of homogeneous ideals in a polynomial ring R. Chapters 2, 3 and 4 deal with homological invariants of edge ideals of bipartite graphs. First, in Chapter 2, we relate regularity and depth of bipartite edge ideals to combinatorial invariants of the graphs. Chapter 3 discusses arithmetic rank, and shows that some classes of Cohen-Macaulay bipartite edge ideals define set-theoretic complete intersections. It is known, due to G. Lyubeznik, that arithmetic rank of a square-free monomial ideal I is at least the pro- jective dimension of R/I. As an application of the results in Chapter 2, we show in Chapter 4 that the multiplicity conjectures of J. Herzog, C. Huneke and H. Srinivasan hold for bipartite edge ideals, and that if the conjectured bounds hold with equality, then the ideals are Cohen-Macaulay and has a pure resolution. Chapter 5 describes joint work with G. Caviglia, showing that any upper bound for projective dimension of an ideal supported on N monomials counted with multiplicity is at least 2N/2. We give the example of a binomial ideal, whose projective dimension grows exponentially with respect to the number of monomials appearing in a set of generators. Finally, in Chapter 6, we study Alexander duality, giving an alternate proof of a theorem of K. Yanagawa which states that for a square-free monomial ideal I, R/I has Serre’s property (Si) if and only if its Alexander dual has a linear resolution up to homological 3 degree i. Further, if R/I has property (S2), then it is locally connected in codimension 1. 4 “... So you’ve won the Scripture-knowledge prize, have you?” “Sir, yes, sir.” “Yes,” said Gussie, “you look just the sort of little tick who would. And yet,” he said, paus- ing and eyeing the child keenly, “how are we to know that this has all been open and above board? Let me test you, G. G. Simmons. What was What’s-His-Name—the chap who begat Thingummy? Can you answer me that, Simmons?” “Sir, no, sir.” Gussie turned to the bearded bloke. “Fishy,” he said. “Very fishy. This boy appears to be totally lacking in Scripture knowledge.” The bearded bloke passed a hand across his forehead. — P. G. Wodehouse, Right Ho, Jeeves 5 Acknowledgments To my advisor, Craig Huneke, I owe much, for teaching me commutative algebra and his guidance during the work that lead to this thesis. It was always a joy to talk to him, not just about doing mathematics and about its professional aspects, but also about Wodehouse. His dedication to students and the care that he shows for good teaching will be a model for my academic life. The thesis committee consisted, additionally, of Perry Alexander, Margaret Bayer, Daniel Katz and Jeremy Martin. I thank for them for their comments on the form and the content of the thesis, and, also, for explaining many aspects of commutative algebra and combinatorics throughout my graduate school years. I also express my gratitude to Giulio Caviglia, Hema Srinivasan and Bernd Ulrich who helped me — apart from Craig Huneke and Daniel Katz — to find a post-doctoral position. A part of this thesis is based on joint work with Giulio Caviglia; I thank him for permission to include it. I have benefited from discussions with many people in commutative algebra and algebraic geometry. Janet Striuli helped me through my first steps in commutative al- gebra. Also valuable was everything that I learnt from Bangere Purnaprajna. I would like to thank David Eisenbud, Srikanth Iyengar, Gennady Lyubeznik, Ezra Miller, Is- abella Novik, Anurag Singh and Irena Swanson for sharing some of their ideas and their encouragement. 6 A rather happy ride through graduate school that I have had owes itself to many people: Garrin Kimmell and Morgan Shortle, Susana Bernad, Andrew Danby, Mirco Speretta, Andy Hom and Halle O’Neil, Nicole Abaid, Jenny Buontempo, Erin Car- mody, Mrinal Das, Aslihan Demirkaya, Tom Enkosky, William Espenschied, Ananth Hariharan, Brandon Humpert, Jeff Mermin, Branden Stone, and Javid Validashti. The support from the Mathematics Department was crucial at every stage. None of this would have been possible without the encouragement and support of my parents. 7 Contents Abstract 3 Acknowledgments 6 Introduction 11 1 Graded Free Resolutions 16 1.1 Graded Modules .............................. 16 1.2 Graded Free Resolutions and Betti Numbers ............... 19 1.3 Monomial Ideals ............................. 24 1.3.1 Initial Ideals ........................... 24 1.3.2 The Taylor Resolution ...................... 25 1.3.3 Polarization ............................ 26 1.3.4 Stanley-Reisner Theory and Alexander Duality ......... 28 2 Regularity and Depth of Bipartite Edge Ideals 33 2.1 Edge Ideals ................................ 33 2.2 Bipartite Edge Ideals ........................... 35 2.3 Examples ................................. 48 2.4 Quasi-pure resolutions .......................... 50 2.5 Discussion ................................. 55 8 3 Arithmetic Rank of Bipartite Edge Ideals 57 3.1 Arithmetic Rank .............................. 57 3.2 Main Result ................................ 59 3.3 Examples ................................. 66 3.4 Further Questions ............................. 67 4 Multiplicity Bounds for Quadratic Monomial Ideals 69 4.1 Introduction ................................ 69 4.2 Earlier work ................................ 71 4.3 Conjectures of Boij-Soderberg¨ ...................... 73 4.4 Some Reductions for Monomial Ideals .................. 74 4.5 Proof of Theorem 4.1.1 .......................... 78 4.6 Proof of Theorem 4.1.2 .......................... 82 5 Monomial Support and Projective Dimension 87 5.1 Monomial Supports ............................ 88 5.2 Main Example ............................... 89 5.3 Further Questions ............................. 94 6 Alexander Duality and Serre’s Property 96 6.1 Introduction ................................ 96 6.2 Free resolutions and the locus of non-(Si) points ............. 98 6.3 Proofs of Theorems ............................ 101 6.4 Discussion ................................. 104 Bibliography 106 List of Notation 111 9 Index 112 10 Introduction Below we present a brief introduction to the problems studied in this dissertation, fol- lowed by main results of each chapter. Detailed discussion of the problems and earlier work is given in each chapter. Main Results The general motif of this dissertation is the study of numerical invariants of free reso- lutions over polynomials rings. Let k be a field and V a finite set of indeterminates. Let R = k[V] and M a finitely generated graded R-module. A graded free resolution of of M is a complex φ2 φ1 (F•,φ•) : ··· / F2 / F1 / F0 / 0 of graded free R-modules Fl such that the homology groups H0(F•) ' M and Hl(F•) = 0 for l > 0. A discussion of graded free resolutions appears in Chapter 1. We review free reso- lutions of monomial ideals in detail, discussing initial ideals, polarization, and Stanley- Reisner theory. In the next three chapters, we look at free resolutions of edge ideals of bipartite graphs. These are ideals generated by quadratic square-free monomials that could be thought of as edges of a bipartite graph. We study ideals correspond- 11 ing to perfectly matched bipartite graphs; this class contains the class of unmixed (and Cohen-Macaulay) ideals. The tool that we use to study the edge ideal of a perfectly matched bipartite graph G is a directed graph dG that we associate to G; see Discussion 2.2.1. We then reformu- late some known results in this framework and give different proofs. For example, in Chapter 2, we give an alternate proof of a result of J. Herzog and T. Hibi characterizing Cohen-Macaulay bipartite graphs. F Theorem 2.2.13. Let G be a bipartite graph on the vertex set V = V1 V2. Then G is Cohen-Macaulay if and only if G is perfectly matched and the associated directed graph dG is acyclic and transitively closed, i.e., it is a poset. We then proceed to study (Castelnuovo-Mumford) regularity and depth of such ide- als. We show that Theorem 2.2.15. Let G be an unmixed bipartite graph with edge ideal I. Then regR/I = max{|A| : A ∈ AdG }. In particular, regR/I = r(I). Here AdG is the set of antichains in dG and r(I) is the maximum size of a pairwise disconnected set of edges. This theorem complements a result of X. Zheng [Zhe04, Theorem 2.18] that if I is the edge ideal of a tree (an acyclic graph) then regR/I = r(I). In Theorem 2.2.17, we describe the resolution of the Alexander dual of a Cohen- Macaulay bipartite edge ideal in terms of the antichains of dG. We use this, along with a result of N. Terai [Ter99] (see [MS05, Theorem 5.59] also), to give a description of the depth of unmixed bipartite edge ideals in Corollary 2.2.18. We conclude Chapter 2 by describing the Cohen-Macaulay bipartite graphs that have quasi-pure resolutions in Proposition 2.4.5; see the opening paragraph in Section 2.4 also. Quasi-pure resolutions are defined on p. 23. The notions of pure and quasi-pure resolutions have appeared in 12 the the multiplicity conjectures of Herzog, C.